Absract: Many counting problems can be modeled by putting objects into boxes. There are several variations of this model, only some of which have been completely solved. This talk will discuss one of the partially solved versions.

September 29,Putnam Club Guest Speaker, 11:00-12:00 p.m., 383 Olin, Robert Rhoades, University of Wisconsin, Madison and 2005Graduate of Bucknell University"All the Tricks that I Know"

Abstract: The Putnam exam is hard. Often it is hard to even start thinking about the problems on the exam. The goal of this talk is to give you some tools that will help to make many of the problems on the Putnam easier. We will assume a basic knowledge of Taylor series.

Abstract: Mathematics is filled with complicated objects. Even worse there are often complicated relationships between the complicated objects in mathematics. One ''complicated'' object is a modular form. In this talk we will relate modular forms to simple combinatorial objects, the binomial coefficients. We will discuss a few major results from the study of modular forms and show how these complicated results boil down to an elegant and simple expression.

Abstract: A function f which satisfies the equation f(x+y) = f(x) + f(y) for all x and y in some space is called a homomorphism. If we consider functions from the real line to itself, it is easy to see that any function of the form f(x)=mx is a homomorphism. We address the question of whether all homomorphisms must be of this form. Using a controversial (yes, controversial) idea from 1905 by Georg Hamel, we show that there exist homomorphisms which are not of this form, and we show that these homomorphisms (in fact, all homomorphisms not of the form f(x)=mx) have the surprising property that their graphs are dense in the plane.

October 14-19Distinguished Visiting Professor, William Kantor: Dept. of Mathematics, University of Oregon

Colloquium - Monday - October 174:00 - OLIN 372

"The Probability of Generating"Abstract. Suppose you write down a couple of square matrices with entries mod 2. If you repeatedly add and multiply like mad, you'll get lots of matrices. What is the set of matrices you'll probably get?Suppose you write down a couple of permutations of {1,2,....n}. If you repeatedly multiply like mad, you'll get lots of permutations. What is the set of permutations you'll probably get?I'll describe the answers, with hints of proofs, starting with elementary tools and working up to the use of properties of all finite simple groups.

An Undergraduate Talk - Tuesday - October 1812:00 - OLIN 372

"SUMS OF SQUARES"Abstract: What integers are sums of two square integers? Three square integers? Four? Which integers can be written as x12+2 x22+5x32+5x42 for some integers x1, x2, x3, x4? I’ll discuss these and related questions, for example by expanding the notion of using both number theory and algebra.

October 27,Student Colloquim Series, 4:00 p.m., ROOKE 116 Auditorium

The Physics Department and the Mathematics Department Present a Joint Student Series Colloquium

Arthur Shapiro, Department of Psychology, Bucknell University

"Perception and Illusion"Visual illusions based on the asynchronous modulation of contrastAbstract: The visual response to color is often thought to be slow (typically, chromatic modulation thresholds at 10 Hz are about eight times higher than at 1 Hz). Recent studies, however, have shown that the visual system can adapt perfectly well to fast chromatic contrast (>10 Hz) and that cells in the primary visual area of the brain respond to fast chromatic contrast modulation. To investigate this apparent discrepancy, my laboratory has developed a class of stimuli based on shifting the temporal phase of multiple sources of contrast information. In this talk, I will show how this class of stimuli can be used to generate a wide variety of compelling visual illusions, and how these illusions lead to new insights into lightness/brightness, color, and motion perception. I will also discuss some simple mathematical models of the phenomena. Examples of the illusions can be seen at www.shapirolab.net.

Bucknell University will be hosting a non-profit mathematics book fair called "Bucknell Math Book Blast" in conjunction with National Book Week on Thursday, November 17, 2005 from 6:00-7:00 PM in the Bertrand Library Traditional Reading Room. Bucknell students will be selling and promoting mathematically related children's books to elementary school teachers, librarians, caregivers and elementary students in addition to presenting related math activities. The book fair is free and open to the public; books will be available for browsing or purchase (cash or check only) and copies of extenstion activities will be provided.

Abstract: A digroup is an algebra defined on a set having two associative binary operations. Digroups play an important role in an open problem from the theory of Leibniz algebras. I will present a brief overview of this problem. I will also discuss some fundamental properties of digroups, including a new axiomatization scheme that I developed as part of an REU research project last summer.

Places I've Been

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